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Subsections

Having calculated the total energy, both the density-kernel and the expansion coefficients for the localised orbitals are varied. Because of the non-orthogonality of the support functions, it is necessary to take note of the tensor properties of the gradients [163], as noted in section 4.6.

## 7.2.1 Density-kernel derivatives

The total energy depends upon both explicitly and through the electronic density . We use the result

 (7.20)

### 7.2.1.1 Kinetic and pseudopotential energies

From equations 7.6, 7.16 and 7.18 we have that

 (7.21)

and therefore
 (7.22)

### 7.2.1.2 Hartree and exchange-correlation energies

The sum of the Hartree and exchange-correlation energies, depends only on the density so that

 (7.23)

The functional derivative of the Hartree-exchange-correlation energy with respect to the electronic density is simply the sum of the Hartree and exchange-correlation potentials, . The electronic density is given in terms of the density-kernel by
 (7.24)

so that we obtain
 (7.25)

Finally, therefore
 (7.26)

### 7.2.1.3 Total energy

Defining the matrix elements of the Kohn-Sham Hamiltonian in the representation of the support functions by

 (7.27)

the derivative of the total energy with respect to the density-kernel is simply
 (7.28)

## 7.2.2 Support function derivatives

Again we can treat the kinetic and pseudopotential energies together, and the Hartree and exchange-correlation energies together. We use the result that

 (7.29)

### 7.2.2.1 Kinetic and pseudopotential energies

We define the kinetic energy operator , whose matrix elements are

 (7.30)

Since the operator is Hermitian,
 (7.31)

Therefore
 (7.32)

The derivation for the pseudopotential energy is identical with the replacement of by the pseudopotential operator, and so the result for the sum of these energies is just
 (7.33)

### 7.2.2.2 Hartree and exchange-correlation energies

Again this gradient is derived by considering the change in the electronic density.

 (7.34)

Therefore
 (7.35)

### 7.2.2.3 Total energy

The gradient of the total energy with respect to changes in the support functions is

 (7.36)

where is the Kohn-Sham Hamiltonian which operates on .

Next: 7.3 Penalty functional and Up: 7. Computational implementation Previous: 7.1 Total energy and   Contents
Peter Haynes